Scalar hairy black holes and scalarons in the isolated horizons formalism

The Isolated Horizons (IH) formalism, together with a simple phenomenological model for colored black holes has been used to predict nontrivial formulas that relate the ADM mass of the solitons and hairy Black Holes of Gravity-Matter system on the one hand, and several horizon properties of the black holes in the other. In this article, the IH formalism is tested numerically for spherically symmetric solutions to an Einstein-Higgs system where hairy black holes were recently found to exist. It is shown that the mass formulas still hold and that, by appropriately extending the current model, one can account for the behavior of the horizon properties of these new solutions. An empirical formula that approximates the ADM mass of hairy solutions is put forward, and some of its properties are analyzed.

Figure 1

Qualitative shape of the scalar-field potential $V(\varphi )$ as given by Eq. (7) used to construct the asymptotically flat black hole and soliton solutions.

Figure 2

The ADM mass (solid lines), the horizon BH mass (dash-dotted lines), and the mass of the Schwarzschild solution (dashed-lines) plotted as functions of $r_{\Delta }$ (first panel). The second panel depicts similar quantities using logarithmic scales to appreciate better their behavior for small $r_{\Delta }$. The soliton mass $M_{\mathrm{sol}}\approx 3.827$.

Figure 3

$\beta (r_{\Delta })=2\kappa r_{\Delta }$ is plotted as function of $r_{\Delta }$. Note that it approaches asymptotically the value $\approx 1.24$. Here $\beta (0)\approx 0.324$.

Figure 4

Large-black-hole configuration constructed with $V(\varphi )$ as given by Eq. (7) with parameters ${\eta }_1=0.5$, ${\eta }_2=0.1$, $a=0$, and $r_{\Delta }=150/\sqrt{\lambda }$, ${\varphi }_{\Delta }\sim 0.26111$. The upper panels depict the scalar field and the mass function, respectively. The latter converges to $M_{\mathrm{ADM}}\sim 93.096/\sqrt{\lambda }$. The lower panels depicts the metric potentials (the first is a zoom of the second): $\sqrt{-g_{tt}}$ (solid line), $\sqrt{g_{rr}}$ (dashed line), $e^{\delta }$ (dash-dotted line) and $\delta$ (dotted line).

Figure 5

Both binding energies are plotted. The “old” binding energy $E_{\mathrm{bind}}(r_{\Delta })$ is shown to become positive and approaches asymptotically a straight line (dashed line). The “new” binding energy $E_{\mathrm{bind}}^{\mathrm{new}}(r_{\Delta })$ is also plotted (solid line), showing the expected behavior.

Figure 6

This figure illustrates a physical process where an initial configuration with an isolated horizon ${\Delta }_{\mathrm{in}}$ is perturbed and the final state contains another isolated horizon ${\Delta }_{\mathrm{fin}}$.

Figure 7

The unstable-mode frequency $\omega =\sqrt{-{\sigma }^2}$ of the perturbed BH and soliton ($r_{\Delta }=0$) is plotted as a function of the horizon radius $r_{\Delta }$. Note that the frequency decreases as the horizon radius increases.

Figure 8

The three panels depict the ADM-mass, $\beta$, and binding energy $E_{\mathrm{bind}}^{\mathrm{new}}$, respectively, as a function of the horizon radius. The solid lines correspond to the values obtained from a numerical analysis and the dashed lines were obtained from the empirical formulas described in the main text. Note the good qualitative behavior of the empirical formulas. Remarkably good fits to the more precise numerical values are obtained for small and large $r_{\Delta }$. The values for the empirical formula are $C\approx 0.62$, $M_{\mathrm{sol}}\approx 3.827$ and ${\beta }_0\approx 0.324$.

Figure 9

Same as Fig. 8, for the Einstein-Yang-Mills theory ($n=1$ colored black holes). Here the values for the empirical formulas are $C=1/2$, $M_{\mathrm{sol}}\approx 0.828$ and ${\beta }_0\approx 0.126$.

Figure 10

Panels 1–4 depict the ADM-mass, horizon mass, $\beta$ and the binding energy, respectively, as a function of $r_{\Delta }$. The solid lines were obtained from the numerical analysis while the dashed lines were computed using the empirical formulas. The lines are associated with the five different values used for ${\eta }_2=0.1$, 0.11, 0.12, 0.13, 0.14 with ${\eta }_1=0.5$ fixed. As seen from bottom to top (for large $r_{\Delta }$) the plots of panels 1–3 correspond to ${\eta }_2$ in increasing order (in panel 4 the order is reversed). The values of the parameters ${\overrightarrow{P}}_{{\eta }_2}=(M_{\mathrm{sol}},C,{\beta }_0{)}_{{\eta }_2}$ used in the empirical formulas are ${\overrightarrow{P}}_{0.1}\approx (3.82,0.62,0.32)$, ${\overrightarrow{P}}_{0.11}\approx (5.22,0.66,0.24)$, ${\overrightarrow{P}}_{0.12}\approx (7.47,0.72,0.17)$ ${\overrightarrow{P}}_{0.13}\approx (11.78,0.82,0.11)$ ${\overrightarrow{P}}_{0.14}\approx (23.51,1.01,0.06)$.